where the symbol \((d{\bf X})\)  stands for the exterior product of the elements of \(d{\bf X}\)[1]}, [2]}, [3]}. In particular, \(\left( {d{\bf {U}}} \right)\)  and \(\left( {d{\bf {\Lambda }}} \right)\)  denote the standard Haar measures of \({{\cal U}\left( {{N_r}} \right)}\)  and \({{\cal U}\left( {{M-1}} \right)}\) , respectively. Hence, by combining (REF ) with the probability density function (PDF) of \(\tilde{\bf Z}_w\) , i.e., \({f_{{{{\bf {\tilde{Z}}}}_w}}}( {{{{\bf {\tilde{Z}}}}_w}} ) = {{{\pi ^{-{N_r}\left( {M - 1} \right)}}}}{\rm {etr}}( { - {{{\bf {\tilde{Z}}}}_w}{{{\bf {\tilde{Z}}}}_w}^{\rm {H}}} )\) , the PDF of \(\bf U\)  is derived as
\({f_{\bf {U}}}\left( {\bf {U}} \right) = \frac{{\left( {M - 1} \right)!\prod \nolimits _{j = 1}^{M - 1} {\left( {{N_r} - j} \right)!\left( {M - 1 - j} \right)!} }}{{\det {{\left( {{{\bf {R}}_r}} \right)}^{M - 1}}{\pi ^{{N_r}\left( {M - 1} \right)}}}}\times \\\int \nolimits _{{\bf \Delta } \in {\mathbb {R}}_ + ^{{N_r} \times \left( {M - 1} \right)}} {{\rm {etr}}\left( { - {{\bf {R}}_r}^{ - 1}{\bf {U}}{\bf \Delta } {{\bf {U}}^{\rm {H}}}} \right)\times }\\\prod \limits _{1 \le i < j \le M - 1} {{{\left( {{\delta _j} - {\delta _i}} \right)}^2}} \prod \limits _{j = 1}^{M - 1} {{\delta _i}^{{N_r} - M + 1}} d{\delta _1} \cdots d{\delta _{M - 1}},\) 
